Shape matching and quantitative comparison of three-dimensional shapes and pharmacophores (a group of atoms in a molecule, having particular properties) between two molecules are main methods in the field of drug design. The volume of a molecule is relevant to its shape which, in turn, determines the physical and chemical properties of the molecule and thus the bioactivity thereof.
Conventionally, a shape of a molecule was calculated by a set of fused hard spheres, where each hard sphere represents an atom with its van der Waals radius. In a hard sphere model, the volume density function F(r) takes a simple value, either 1 if the coordinate r is within the molecule or 0 if r is outside. Therefore, the volume density function can be expressed in terms of the volume density functions of individual atoms as Formula (1):
                                                                        F                ⁡                                  (                  r                  )                                            =                            ⁢                                                                    ∑                    i                                    ⁢                                                            f                      i                                        ⁡                                          (                      r                      )                                                                      -                                                      ∑                                          i                      <                      j                                                        ⁢                                                                                    f                        i                                            ⁡                                              (                        r                        )                                                              ⁢                                                                  f                        j                                            ⁡                                              (                        r                        )                                                                                            +                                                      ∑                                          i                      <                      j                      <                      k                                                        ⁢                                                                                    f                        i                                            ⁡                                              (                        r                        )                                                              ⁢                                                                  f                        j                                            ⁡                                              (                        r                        )                                                              ⁢                                                                  f                        k                                            ⁡                                              (                        r                        )                                                                                            -                                                                                                      ⁢                                                                    ∑                                          i                      <                      j                      <                      k                      <                      l                                                        ⁢                                                                                    f                        i                                            ⁡                                              (                        r                        )                                                              ⁢                                                                  f                        j                                            ⁡                                              (                        r                        )                                                              ⁢                                                                  f                        k                                            ⁡                                              (                        r                        )                                                              ⁢                                                                  f                        l                                            ⁡                                              (                        r                        )                                                                                            +                …                                                                                        =                            ⁢                              1                -                                                      ∏                    i                                    ⁢                                                                          ⁢                                      [                                          1                      -                                                                        f                          i                                                ⁡                                                  (                          r                          )                                                                                      ]                                                                                                          (        1        )            wherein fi(r) represents the volume density of atom i.
As shown in FIG. 1, for a diatomic molecule, the volume is calculated by subtracting overlap of the two atoms from the summation volume, i.e, V=VA+VB−VAB. For a triatomic molecule, the volume is calculated by alternative “inclusion” and “exclusion” terms: V=VA+VB+VC−VAB−VAC−VBC+VABC. For any molecules, the volume is also calculated by alternative “inclusion” and “exclusion” terms, taking advantage of Formula (2):
                                                        V              =                            ⁢                              ∫                                                      F                    ⁡                                          (                      r                      )                                                        ⁢                  dr                                                                                                        =                            ⁢                                                                    ∑                    i                                    ⁢                                      v                    i                                                  -                                                      ∑                                          i                      <                      j                                                        ⁢                                      v                    ij                                                  +                                                      ∑                                          i                      <                      j                      <                      k                                                        ⁢                                      v                    ijk                                                  -                                                      ∑                                          i                      <                      j                      <                      k                      <                      l                                                        ⁢                                      v                    ijkl                                                  +                …                                                                        (        2        )            wherein vi represents the volume of atom i, vij represents the (second order) overlap of atoms i and j, vijk represents the (third order) overlap of atoms i, j and k, and so on. For a molecule having N atoms, the summation generally goes on up to the order of N−1.
One of the main disadvantages of the molecular shape calculation method is prolonged calculation time and decreased accuracy due to the discontinuity of the derivative of the volume obtained through the hard sphere model with respect to the atom coordinate.